source: sasmodels/sasmodels/models/flexible_cylinder_ex.py @ d507c3a

r"""
This model calculates the form factor for a flexible cylinder with an
elliptical cross section and a uniform scattering length density.
The non-negligible diameter of the cylinder is included by accounting
for excluded volume interactions within the walk of a single cylinder.
The form factor is normalized by the particle volume such that

.. math::

    P(q) = \text{scale} \left<F^2\right>/V + \text{background}

where the averaging $\left<\ldots\right>$ is over all possible orientations
of the flexible cylinder.

The 2D scattering intensity is the same as 1D, regardless of the orientation
of the q vector which is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}


Definitions
-----------

The function calculated is from the reference given below. From that paper,
"Method 3 With Excluded Volume" is used.
The model is a parameterization of simulations of a discrete representation of
the worm-like chain model of Kratky and Porod applied in the pseudo-continuous
limit. See equations (13, 26-27) in the original reference for the details.

.. note::

    There are several typos in the original reference that have been corrected
    by WRC. Details of the corrections are in the reference below. Most notably

    - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$

    - Equations (23) and (24) are incorrect; WRC has entered these into
      Mathematica and solved analytically. The results were then converted to
      code.

    - Equation (27) should be $q0 = max(a3/sqrt(RgSquare),3)$ instead of
      $max(a3*b/sqrt(RgSquare),3)$

    - The scattering function is negative for a range of parameter values and
      q-values that are experimentally accessible. A correction function has been
      added to give the proper behavior.

.. figure:: img/flexible_cylinder_ex_geometry.jpg


The chain of contour length, $L$, (the total length) can be described as a chain
of some number of locally stiff segments of length $l_p$, the persistence length
(the length along the cylinder over which the flexible cylinder can be considered
a rigid rod).
The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.

The cross section of the cylinder is elliptical, with minor radius $a$ .
The major radius is larger, so of course, **the axis ratio (parameter 5) must be
greater than one.** Simple constraints should be applied during curve fitting to
maintain this inequality.

The returned value is in units of $cm^-1$, on absolute scale.

In the parameters, the $sldCyl$ and $sldSolv$ represent the SLD of the
chain/cylinder and solvent respectively. The *scale*, and the contrast are both
multiplicative factors in the model and are perfectly correlated. One or both of
these parameters must be held fixed during model fitting.

**No inter-cylinder interference effects are included in this calculation.**

.. figure:: img/flexible_cylinder_ex_1d.jpg

    1D plot using the default values (w/1000 data point).

References
----------

J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers
with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612

Correction of the formula can be found in

W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in
the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir,
22(15) 2006 6539-6548
"""
from numpy import inf

name = "flexible_cylinder_ex"
title = "Flexible cylinder wth an elliptical cross section and a uniform " \
        "scattering length density."
description = """Note : scale and contrast=sldCyl-sldSolv are both multiplicative
        factors in the
        model and are perfectly correlated. One or
        both of these parameters must be held fixed
        during model fitting.
        """
single = False

category = "shape:cylinder"
# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
    ["length",      "Ang",       1000.0, [0, inf],    "volume", "Length of the flexible cylinder"],
    ["kuhn_length", "Ang",        100.0, [0, inf],    "volume", "Kuhn length of the flexible cylinder"],
    ["radius",      "Ang",         20.0, [0, inf],    "volume", "Radius of the flexible cylinder"],
    ["axis_ratio",  "",             1.5, [0, inf],    "",       "Axis_ratio (major_radius/radius"],
    ["sld",         "1e-6/Ang^2",   1.0, [-inf, inf], "",       "Cylinder scattering length density"],
    ["solvent_sld", "1e-6/Ang^2",   6.3, [-inf, inf], "",       "Solvent scattering length density"],
    ]
# pylint: enable=bad-whitespace, line-too-long

source = ["lib/J1.c", "lib/gauss76.c", "lib/wrc_cyl.c", "flexible_cylinder_ex.c"]

demo = dict(scale=1.0, background=0.0001,
            length=1000.0,
            kuhn_length=100.0,
            radius=20.0,
            axis_ratio=1.5,
            sld=1.0,
            solvent_sld=6.3)

oldname = 'FlexCylEllipXModel'
oldpars = dict(sld='sldCyl', solvent_sld='sldSolv')


tests = [
    # Accuracy tests based on content in test/utest_other_models.py
    # Currently fails in OCL
    # [{'length':     1000.0,
    #  'kuhn_length': 100.0,
    #  'radius':       20.0,
    #  'axis_ratio':    1.5,
    #  'sld':           1.0,
    #  'solvent_sld':   6.3,
    #  'background':    0.0001,
    # }, 0.001, 3509.2187],

    # Additional tests with larger range of parameters
    [{'length':     1000.0,
      'kuhn_length': 100.0,
      'radius':       20.0,
      'axis_ratio':    1.5,
      'sld':           1.0,
      'solvent_sld':   6.3,
      'background':    0.0001,
     }, 1.0, 0.00223819],
    [{'length':        10.0,
      'kuhn_length': 800.0,
      'radius':        2.0,
      'axis_ratio':    0.5,
      'sld':           6.0,
      'solvent_sld':  12.3,
      'background':    0.001,
     }, 0.1, 0.390281],
    [{'length':        100.0,
      'kuhn_length': 800.0,
      'radius':       50.0,
      'axis_ratio':    4.5,
      'sld':           0.1,
      'solvent_sld':   5.1,
      'background':    0.0,
     }, 1.0, 0.0016338264790]
    ]